1,354,834 research outputs found
A variational formulation approach to a generalized coupled inhomogeneous Emden–Fowler system
No full textWe study a generalized coupled inhomogeneous Emden–Fowler system from the Lagrangian formulation standpoint. A special case of this system was considered in the literature, and necessary and sufficient conditions for the existence of multiple positive solutions were obtained. Here we perform preliminary Noether classification of the generalized system by the direct method. We obtain nine cases for which the system has Noether point symmetries. First integrals are then obtained for the cases which admit Noether point symmetries
On the solutions and conservation laws for the Sharma–Tasso–Olver equation
No full textWe study the Sharma-Tasso-Olver equation from the Lie symmetry point of view. We derive the Lie point symmetry generators of the equation and classify them to obtain the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the equation. These subalgebras are then used to construct symmetry reductions for the equation. We obtain the general solution of the nonlinear second-order ordinary differential equation which results from the symmetry reduction for the travelling wave group-invariant solutions of the equation by transforming it into a linear third-order ordinary differential equation through a Riccati transformation. Then we show that one can easily obtain the travelling wave exact group-invariant solutions for the underlying equation by using the general solution of the linearized third-order ordinary differential equation and the Riccati transformation. We also construct conservation laws for the underlying equation by making use of the multiplier method
Exact Solitary Wave and Periodic Wave Solutions of a Class of Higher–Order Nonlinear Wave Equations
No full textWe study the exact traveling wave solutions of a general fifth-order nonlinear wave equation and a generalized sixth-order KdV equation. We find the solvable lower-order subequations of a general related fourth-order ordinary differential equation involving only even order derivatives and polynomial functions of the dependent variable. It is shown that the exact solitary wave and periodic wave solutions of some high-order nonlinear wave equations can be obtained easily by using this algorithm. As examples, we derive some solitary wave and periodic wave solutions of the Lax equation, the Ito equation, and a general sixth-order KdV equation
Nonlinearly Self-Adjoint, Conservation Laws and Solutions for a Forced BBM Equation
No full textWe study a forced Benjamin-Bona-Mahony (BBM) equation. We prove that the equation is not weak self-adjoint; however, it is nonlinearly self-adjoint. By using a general theorem on conservation laws due to Nail Ibragimov and the symmetry generators, we find conservation laws for these partial differential equations without classical Lagrangians. We also present some exact solutions for a special case of the equation
Group Analysis of a Generalized KdV Equation
No full textIn this work the Korteweg-de Vries equation which contains an arbitrary function in the nonlinear term is considered and it is referred to as a generalized KdV. This equation has applications in nonlinear solitary wave phenomena in some areas of fluid mechanics, plasma physics and quantum mechanics. The Lie group analysis approach is employed to obtain the possible forms of the arbitrary parameter
Conservation laws and solutions of a generalized coupled (2+1)-dimensional Burgers system
No full textIn this paper we study a generalized coupled (2+1)-dimensional Burgers system, which is a nonlinear version of a bilinear system under some dependent variable transformations. It was introduced recently in the literature and has attracted a fair amount of interest from physicists. The Lie symmetry analysis together with the Kudryashov approach are utilized to obtain new travelling wave solutions of the system. Furthermore, for the first time, conservation laws of the system are derived using the multiplier method
Exact solutions for Stokes' Flow of a non-Newtonian nanofluid model: a Lie Similarity approach
No full textThe fully developed time-dependent flow of an incompressible, thermodynamically compatible non-Newtonian third-grade nanofluid is investigated. The classical Stokes model is considered in which the flow is generated due to the motion of the plate in its own plane with an impulsive velocity. The Lie symmetry approach is utilised to convert the governing nonlinear partial differential equation into different linear and nonlinear ordinary differential equations. The reduced ordinary differential equations are then solved by using the compatibility and generalised group method. Exact solutions for the model equation are deduced in the form of closed-form exponential functions which are not available in the literature before. In addition, we also derived the conservation laws associated with the governing model. Finally, the physical features of the pertinent parameters are discussed in detail through several graphs
Direct approach to a group classification problem: Fisher equation with time-dependent coefficients
No full textWe perform Lie symmetry analysis of a time-variable coefficient Fisher equation which models reaction–diffusion–convection phenomena in biological, chemical and physical systems. These time-dependent coefficients (model parameters or arbitrary elements) are specified via the direct integration of the classifying relations
Symmetries, solutions and conservation laws of a class of nonlinear dispersive wave equations
No full textIn this paper we consider a damped externally excited Korteweg-de Vries (KdV) equation with a forcing term. We derive the classical Lie symmetries admitted by the equation. We then find that the damped externally excited KdV equation has some exact solutions which are periodic waves and solitary waves. These solutions are derived from the solutions of a simple nonlinear ordinary differential equation. By using a general theorem on conservation laws and the multiplier method, we construct some conservation laws for some of these partial differential equations
Conversation laws for a variable coefficient variant boussinesq system
No full textWe construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.http://www.hindawi.com/journals/aaa/2014/169694/http://dx.doi.org/10.1155/2014/16969
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